Linearized Research Articles

STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

Abstract:
 In this paper we study the stability of one of a non linear autoregressive model with trigonometric term by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude that the proposed model under certain conditions have a non-zero singular point which is a asymptotically salable ( when 0 ) and have an orbitaly stable limit cycle . Also we give some examples in order to explain the method.
 Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

Implicit fast sweeping method for hyperbolic systems of conservation laws

Implicit time-accurate methods are often used to integrate stiff problems where explicit schemes impose severe time step restrictions. This paper presents an efficient numerical framework based on the fast sweeping method (FSM) for solving linear and nonlinear hyperbolic systems of conservation laws. The solution at each discrete location is computed by sweeping the numerical domain in several predetermined directions that follow the causality of the characteristic families. The use of a fractional step strategy eliminates the need for a solution selection criterion while one-sided stencils limit the number of sweeps to at most 2d for d space dimensions. This work focuses on the first-order implicit upwind method since it constitutes the building block for high-order conservative schemes. For problems where the degree of stiffness evolves over time, implicit-explicit hybridization can be accomplished with the same algorithm by simply switching the stencil at each time level. As opposed to traditional implicit solvers, the sweeping method does not require a local time linearization of the fluxes thereby preserving the nonlinear stability properties of the original implicit scheme. It also avoids the large computational and memory requirements associated with solving large block-diagonal systems of equations. A series of one- and two-dimensional test cases are presented for the inviscid Burgers' equation and the reactive Euler equations. The results indicate that the implicit FSM can allow for a major reduction in the number of time steps even in the presence of discontinuous solution profiles.

Cross-covariance based affinity for graphs.

The accuracy of graph based learning techniques relies on the underlying topological structure and affinity between data points, which are assumed to lie on a smooth Riemannian manifold. However, the assumption of local linearity in a neighborhood does not always hold true. Hence, the Euclidean distance based affinity that determines the graph edges may fail to represent the true connectivity strength between data points. Moreover, the affinity between data points is influenced by the distribution of the data around them and must be considered in the affinity measure. In this paper, we propose two techniques, CCGAL and CCGAN that use cross-covariance based graph affinity (CCGA) to represent the relation between data points in a local region. CCGAL also explores the additional connectivity between data points which share a common local neighborhood. CCGAN considers the influence of respective neighborhoods of the two immediately connected data points, which further enhance the affinity measure. Experimental results of manifold learning on synthetic datasets show that CCGA is able to represent the affinity measure between data points more accurately. This results in better low dimensional representation. Manifold regularization experiments on standard image dataset further indicate that the proposed CCGA based affinity is able to accurately identify and include the influence of the data points and its common neighborhood that increase the classification accuracy. The proposed method outperforms the existing state-of-the-art manifold regularization methods by a significant margin.

Stability Regions of Vehicle Lateral Dynamics: Estimation and Analysis

Abstract A new method is proposed to estimate and analyze the vehicle lateral stability region, which provides a direct and intuitive demonstration for the safety and stability control of ground vehicles. Based on a four-wheel vehicle model and a nonlinear two-dimensional (2D) analytical LuGre tire model, a local linearization method is applied to estimate the vehicle lateral stability regions by analyzing the vehicle stability at each operation point on a phase plane, which includes but not limited to the equilibrium points. As the collections of all the locally stable operation points, the estimated stability regions are conservative because both vehicle and tire stability are simultaneously considered, which are especially important for characterizing the stability features of highly/fully automated ground vehicles (AGV). The obtained lateral stability regions can be well explained by the vehicle characteristics of oversteering and understeering in the context of vehicle handling stability. The impacts of vehicle lateral load transfer, longitudinal velocity, tire-road friction coefficient, and steering angle on the estimated stability regions are presented and discussed. To validate the correctness of the estimated stability regions, a case study by matlab/simulink and CarSim® co-simulation is presented and discussed.

A geostatistical Markov chain Monte Carlo inversion algorithm for electrical resistivity tomography

ABSTRACTElectrical resistivity tomography is an ill‐posed and nonlinear inverse problem commonly solved through deterministic gradient‐based methods. These methods guarantee a fast convergence towards the final solution, but the local linearization of the inverse operator impedes accurate uncertainty assessments. On the contrary, numerical Markov chain Monte Carlo algorithms allow for accurate uncertainty appraisals, but appropriate Markov chain Monte Carlo recipes are needed to reduce the computational effort and make these approaches suitable to be applied to field data. A key aspect of any probabilistic inversion is the definition of an appropriate prior distribution of the model parameters that can also incorporate spatial constraints to mitigate the ill conditioning of the inverse problem. Usually, Gaussian priors oversimplify the actual distribution of the model parameters that often exhibit multimodality due to the presence of multiple litho‐fluid facies. In this work, we develop a novel probabilistic Markov chain Monte Carlo approach for inversion of electrical resistivity tomography data. This approach jointly estimates resistivity values, litho‐fluid facies, along with the associated uncertainties from the measured apparent resistivity pseudosection. In our approach, the unknown parameters include the facies model as well as the continuous resistivity values. At each spatial location, the distribution of the resistivity value is assumed to be multimodal and non‐parametric with as many modes as the number of facies. An advanced Markov chain Monte Carlo algorithm (the differential evolution Markov chain) is used to efficiently sample the posterior density in a high‐dimensional parameter space. A Gaussian variogram model and a first‐order Markov chain respectively account for the lateral continuity of the continuous and discrete model properties, thereby reducing the null‐space of solutions. The direct sequential simulation geostatistical method allows the generation of sampled models that honour both the assumed marginal prior and spatial constraints. During the Markov chain Monte Carlo walk, we iteratively sample the facies, by moving from one mode to another, and the resistivity values, by sampling within the same mode. The proposed method is first validated by inverting the data calculated from synthetic models. Then, it is applied to field data and benchmarked against a standard local inversion algorithm. Our experiments prove that the proposed Markov chain Monte Carlo inversion retrieves reliable estimations and accurate uncertainty quantifications with a reasonable computational effort.

Numerical analysis of activation energy on MHD nanofluid flow with exponential temperature-dependent viscosity past a porous plate

The current investigation scrutinized the incompressible steady flow with temperature-dependent viscosity of magnetohydrodynamics nanofluid through a vertically stretched porous sheet. The Reynolds exponential model is employed to formulate the mathematical modeling. The momentum equation is further devised utilizing the Darcy–Brinkman–Forchheimer model. Electrically conducting nanofluids encompass uniformly suspended nanoparticles in the viscous base fluid. The Buongiorno model is adopted that aspects the behavior of thermophoretic forces and Brownian motion. The momentum, mass conservative, energy, and nanoparticle concentration equations are defined with magnetic body force. The looming nonlinear coupled differential equations are resolved numerically by employing the spectral local linearization method (SLLM). The SLLM algorithm is straightforward to develop and apply, as it is based on a smooth univariant linearization of nonlinear functions. The numerical performance of SLLM is more impressive as it grows a set of equations; those are successively solved by operating the results from the one equation into the subsequent equation. To accelerate and improve the convergence for the SLLM scheme, the successive over relaxation scheme has been utilized. The accuracy of the SLLM will be confirmed through the known methods, and convergence analysis is also presented. Graphical conduct for all the emerging parameters across temperature, velocity, and concentration distributions, as well as the Nusselt number, skin friction, and Sherwood number, is presented and discussed in detail. A comparative study of the novel proposed technique along with the preceding explored literature is also granted. It is costly to affirm that the spectral local linearization scheme is uncovered to be much stable and adaptable to solve the nonlinear problems.

Effect of dispersion on thermally stable stratified power‐law fluids over the vertical frustum of a cone in a non‐Darcy porous medium: Flow separation

AbstractThe present problem shares out the influences of convective heating and thermal dispersion on thermally stable stratified power‐law fluid flow over the vertical frustum of a cone situated in a non‐Darcy porous medium. For the first time, it is considered in both aiding and opposing flow cases as it is readily applicable in realistic, practical situations. The local nonsimilarity approach, along with the spectral local linearization method, is used to analyze the fluid behavior and heat transfer within the boundary layer region. In addition, error analysis and comparison with the existing results are also included to validate the obtained results wherever feasible. It is significant to perceive the existence of flow separation in this study as more gain of the velocity of a power‐law fluid is seen for aiding flow case in comparison with the opposing flow case. Also, it is evident from these results that the separation of flow is less in the presence of thermal stratification, but it is more in the absence of thermal stratification. Finally, thermal stratification significantly affects the heat transfer rate of power‐law fluids besides delaying the boundary layer separation.

Tamper-resistant controller using neural network and time-varying quantization

In this paper, we consider a tamper-resistant control system aiming at protecting the knowledge of the controller from attackers. In this control system, the controller operates normally only for a limited number of time-varying specific states; otherwise, it outputs an incorrect value. We propose to realize the tamper-resistant controller by employing a neural network and time-varying quantization. Furthermore, we make it possible for only one trained neural network to be used for all quantization based on the local approximation linearity of the trained neural network. Without this approach, the neural network needs to be trained for every possible quantization, which leads to huge computation. We provide simulations to demonstrate the security and feasibility of the proposed method.

Identification the number of Mycobacterium tuberculosis based on sputum image using local linear estimator

Infectious disease caused by infection of Mycobacterium tuberculosis is called tuberculosis (TB). A common method in detecting TB is by identifying number of mycobacterium TB in sputum manually. Unfortunately, manually calculation by pathologists take a relatively long time. Previous researches on TB bacteria were still limited to detect the absence or presence of mycobacterium TB in images of sputum. This research aims are identifying number of mycobacterium TB and determining accuracy of classification TB severity by approaching nonparametric Poisson regression model and applying an estimator namely local linear. Steps include processing of image, reducing of dimension by applying partial least square and discrete wavelet transformation, and then identifying the number of mycobacterium TB by using the proposed model approach. In this research, we get deviance values of 28.410 for nonparametric and 93.029 for parametric approaches and the average of classification accuracy values for 4 iterations of 92.75% for nonparametric and 85.5% for parametric approaches. Thus, for identifying many of mycobacterium TB met in images of sputum and classifying of TB severity, the proposed identifying method gives higher accuracy and shorter time in identifying number of mycobacterium TB than parametric linear regression method.

Elzaki Substitution Method for Solving Nonlinear Partial Differential Equations with Mixed Partial Derivatives Using Adomian Polynomial

In this paper we apply a new method, named Elzaki Substitution Method to solve nonlinear homogeneous and nonhomogeneous partial differential equations with mixed partial derivatives, which is based on Elzaki Transform. The proposed method introduces also Adomian polynomials and the nonlinear terms can be handled by the use of this polynomials. The proposed method worked perfectly to find the exact solutions of partial equations with mixed partial derivatives without any need of linearization or discretization in comparison with other methods such as Method of Separation of Variables (MSV) and Variation Iteration Method (VIM). Some illustrative examples are given to demonstrate the applicability and efficiency of proposed method.

Reweighted Nadaraya–Watson estimation of conditional density function in the right-censored model

For a conditional density function (CDF) in the right-censored model, the local linear (LL) estimator has superior bias properties compared with the Nadaraya–Watson (NW) one, but it may have negative values and thus give rise to unsuitable inference. In order to alleviate the possible negativity of the LL estimator, we define a reweighted NW (RNW) estimator of the CDF in the right-censored model by employing the empirical likelihood (EL) method. The RNW estimator is constructed by modifying the NW estimator slightly, so it naturally inherits the nonnegativity of the NW one. It is assumed that the censoring time is independent of the survival time with the associated covariate. Under stationary α−mixing observations, the weak consistency and asymptotic normality of the RNW estimator are developed. The asymptotic normality shows that the RNW estimator possesses the bias and variance of the LL estimator. Finally, we conduct simulations to evaluate the finite sample performance of the estimator.

Spectral local linearisation method for MHD Casson fluid on stratified bioconvective porous medium flow due to gyrotactic microorganisms

AbstractIn this paper, the effect of hall parameter on the flow of Casson nanofluid containing gyrotactic microorganisms over a stretching boundary in a porous medium is studied. The stratification, porosity, and Casson fluid parameters are also examined. Using suitable similarity transformations, the basic equations describing the flow are converted to nonlinear differential equations, which are then solved computationally using the spectral local linearisation method. The effects of key parameters such as the Hall parameter, the thermophoresis parameter, the porosity parameter, and Casson fluid parameters are analyzed. The results obtained suggest that the Hall parameter has the effect of decreasing the secondary flow, the heat and mass transfer rate, and density of the motile microorganisms. A decrease in the the Hall parameter is found to cause an increase in the transfer rate, the mass transfer rate, and the density of the motile microorganisms. An increase in the porosity parameter leads to a decline in the skin friction, heat transfer rate, mass transfer rate, and density of the motile microorganisms. The applications of this study arise in industrial areas, including Hall current accelerators, planetary dynamics, Hall current sensors, and magnetohydrodynamic power generators.

Cycle-to-Cycle Multipulse Fuel-Injection Control for Advanced Diesel Combustion in Changing and Disturbed Operating Points

Abstract When controlled in an open-loop fashion, both stationary and dynamical diesel engine performance can deteriorate due to disturbances. Although cycle-to-cycle fuel injection control has the potential of reducing the performance degradation by manipulating the fueling profile in real time, this potential has not been thoroughly studied. This is especially true for multipulse fuel injection, due to the lack of a controller design method with closed-loop reference tracking performance guarantees. In this paper, we introduce a systematic design method for multivariable cycle-to-cycle fuel injection control. The design method is based on several local linearizations to address varying combustion behavior due to different engine speeds, torques, and disturbances. This leads to a controller that is robust with respect to all the local linearizations. Experimental results show that the controller works well in both stationary and dynamical operating conditions, and has fast dynamical performance (settling time less than 1.5 (s)). Moreover, experimental data show that cycle-to-cycle fuel injection control is robust against the uncertainties introduced by disturbed rail pressure and swirl valve position.

AN APPROACH ON ADVANCED UNSCENTED KALMAN FILTER FROM MOBILE ROBOT-SLAM

Abstract. In the past 30 years, Kalman filter is a classical method to solve the problem of simultaneous localization and mapping (SLAM) of mobile robots. Extended Kalman filter (EKF) and unscented Kalman filter (UKF) are derived from Kalman filter. Extended Kalman filter (EKF) overcomes the nonlinear problem that Kalman filter cannot solve. However, when it is strongly nonlinear, EKF violates the assumption of local linearity, and EKF algorithm may make the filter diverge. Secondly, because the Jacobian matrix is needed in the online processing of EKF, its tedious calculation process makes the implementation of this method relatively difficult. Unscented Kalman filter (UKF) uses nonlinear model directly, avoids operation of Jacobian matrix of complex nonlinear function, and ensures the general adaptability of nonlinear system. In this paper, based on the square root unscented Kalman filter, sigma points are selected according to the square-root decomposition of prior covariance, and then weighted mean and covariance are calculated. The quaternion is used to represent the attitude, and the quaternion vector is converted to the rotation space for matrix operation. Comparing the robot poses estimated based on the square root traceless Kalman filter (QSR-UKF), square root traceless Kalman filter (SR-UKF), and extended Kalman filter (EKF), the simulation results show the QSR-UKF proposed in this paper is effective.

A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model

Regions detection has an influence on the better treatment of brain tumors. Existing algorithms in the early detection of tumors are difficult to diagnose reliably. In this paper, we introduced a new robust algorithm using three methods for the classification of brain disease. The first method is Wavelet-Generalized Autoregressive Conditional Heteroscedasticity-K-Nearest Neighbor (W-GARCH-KNN). The Two-Dimensional Discrete Wavelet (2D-DWT) is utilized as the input images. The sub-banded wavelet coefficients are modeled using the GARCH model. The features of the GARCH model are considered as the main property vector. The second method is the Developed Wavelet-GARCH-KNN (D-WGK), which solves the incompatibility of the WGK method for the use of a low pass sub-band. The third method is the Wavelet Local Linear Approximation (LLA)-KNN, which we used for modeling the wavelet sub-bands. The extracted features were applied separately to determine the normal image or brain tumor based on classification methods. The classification was performed for the diagnosis of tumor types. The empirical results showed that the proposed algorithm obtained a high rate of classification and better practices than recently introduced algorithms while requiring a smaller number of classification features. According to the results, the Low-Low sub-bands are not adopted with the GARCH model; therefore, with the use of homomorphic filtering, this limitation is overcome. The results showed that the presented Local Linear (LL) method was better than the GARCH model for modeling wavelet sub-bands.

Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. This new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2020 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.

Adaptive iterative linearization Galerkin methods for nonlinear problems

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed.

Stable control of MSCMG rotor based on feedback linearization

Stable control of MSCMG rotor based on feedback linearization

Stabilizing Control of an Unmanned Surface Vehicle Pushing a Floating Load

In this paper, the automatic control of a single unmanned surface vehicle (USV) pushing a floating load is developed and theoretically analyzed. This represents a challenging control problem, since the manipulated load is underactuated and its open-loop dynamics is inherently unstable. Thus, a stabilizing controller must be designed. To this end, a scheme that combines partial feedback linearization with local linearization of the remaining nonlinear terms is proposed. Such scheme simplifies the design of a variable structure controller, that has robustness characteristics to parametric uncertainties and matched disturbances. The proposed closed-loop control system has local stability properties. Small-scale experimental results in calm waters, and simulation results, illustrate the performance of the proposed control system.

On local linearization method for stochastic differential equations driven by fractional Brownian motion

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fractional linear stochastic differential equation, whose solution can be figured out explicitly. As in the Brownian motion case (i.e., H = 1/2), the rate of convergence, in our case, is twice the one of the Euler scheme. Numerical examples are given to demonstrate the performance of the method.